Polytope of Type {68,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {68,6}*1632
if this polytope has a name.
Group : SmallGroup(1632,1195)
Rank : 3
Schlafli Type : {68,6}
Number of vertices, edges, etc : 136, 408, 12
Order of s0s1s2 : 102
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {68,6}*816b
   4-fold quotients : {34,6}*408
   12-fold quotients : {34,2}*136
   17-fold quotients : {4,6}*96
   24-fold quotients : {17,2}*68
   34-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   68-fold quotients : {4,3}*24, {2,6}*24
   136-fold quotients : {2,3}*12
   204-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      8 facets:
         4 of {34}*68
         4 of {68}*136
      68 vertex figures:
         68 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      6 facets:
         6 of {68}*136
      68 vertex figures:
         68 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,  3)(  2,  4)(  5, 67)(  6, 68)(  7, 65)(  8, 66)(  9, 63)( 10, 64)( 11, 61)( 12, 62)( 13, 59)( 14, 60)( 15, 57)( 16, 58)( 17, 55)( 18, 56)( 19, 53)( 20, 54)( 21, 51)( 22, 52)( 23, 49)( 24, 50)( 25, 47)( 26, 48)( 27, 45)( 28, 46)( 29, 43)( 30, 44)( 31, 41)( 32, 42)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 69, 71)( 70, 72)( 73,135)( 74,136)( 75,133)( 76,134)( 77,131)( 78,132)( 79,129)( 80,130)( 81,127)( 82,128)( 83,125)( 84,126)( 85,123)( 86,124)( 87,121)( 88,122)( 89,119)( 90,120)( 91,117)( 92,118)( 93,115)( 94,116)( 95,113)( 96,114)( 97,111)( 98,112)( 99,109)(100,110)(101,107)(102,108)(103,105)(104,106)(137,139)(138,140)(141,203)(142,204)(143,201)(144,202)(145,199)(146,200)(147,197)(148,198)(149,195)(150,196)(151,193)(152,194)(153,191)(154,192)(155,189)(156,190)(157,187)(158,188)(159,185)(160,186)(161,183)(162,184)(163,181)(164,182)(165,179)(166,180)(167,177)(168,178)(169,175)(170,176)(171,173)(172,174)(205,207)(206,208)(209,271)(210,272)(211,269)(212,270)(213,267)(214,268)(215,265)(216,266)(217,263)(218,264)(219,261)(220,262)(221,259)(222,260)(223,257)(224,258)(225,255)(226,256)(227,253)(228,254)(229,251)(230,252)(231,249)(232,250)(233,247)(234,248)(235,245)(236,246)(237,243)(238,244)(239,241)(240,242)(273,275)(274,276)(277,339)(278,340)(279,337)(280,338)(281,335)(282,336)(283,333)(284,334)(285,331)(286,332)(287,329)(288,330)(289,327)(290,328)(291,325)(292,326)(293,323)(294,324)(295,321)(296,322)(297,319)(298,320)(299,317)(300,318)(301,315)(302,316)(303,313)(304,314)(305,311)(306,312)(307,309)(308,310)(341,343)(342,344)(345,407)(346,408)(347,405)(348,406)(349,403)(350,404)(351,401)(352,402)(353,399)(354,400)(355,397)(356,398)(357,395)(358,396)(359,393)(360,394)(361,391)(362,392)(363,389)(364,390)(365,387)(366,388)(367,385)(368,386)(369,383)(370,384)(371,381)(372,382)(373,379)(374,380)(375,377)(376,378);;
s1 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 65)( 10, 66)( 11, 68)( 12, 67)( 13, 61)( 14, 62)( 15, 64)( 16, 63)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)( 24, 55)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 33, 41)( 34, 42)( 35, 44)( 36, 43)( 39, 40)( 69,141)( 70,142)( 71,144)( 72,143)( 73,137)( 74,138)( 75,140)( 76,139)( 77,201)( 78,202)( 79,204)( 80,203)( 81,197)( 82,198)( 83,200)( 84,199)( 85,193)( 86,194)( 87,196)( 88,195)( 89,189)( 90,190)( 91,192)( 92,191)( 93,185)( 94,186)( 95,188)( 96,187)( 97,181)( 98,182)( 99,184)(100,183)(101,177)(102,178)(103,180)(104,179)(105,173)(106,174)(107,176)(108,175)(109,169)(110,170)(111,172)(112,171)(113,165)(114,166)(115,168)(116,167)(117,161)(118,162)(119,164)(120,163)(121,157)(122,158)(123,160)(124,159)(125,153)(126,154)(127,156)(128,155)(129,149)(130,150)(131,152)(132,151)(133,145)(134,146)(135,148)(136,147)(205,209)(206,210)(207,212)(208,211)(213,269)(214,270)(215,272)(216,271)(217,265)(218,266)(219,268)(220,267)(221,261)(222,262)(223,264)(224,263)(225,257)(226,258)(227,260)(228,259)(229,253)(230,254)(231,256)(232,255)(233,249)(234,250)(235,252)(236,251)(237,245)(238,246)(239,248)(240,247)(243,244)(273,345)(274,346)(275,348)(276,347)(277,341)(278,342)(279,344)(280,343)(281,405)(282,406)(283,408)(284,407)(285,401)(286,402)(287,404)(288,403)(289,397)(290,398)(291,400)(292,399)(293,393)(294,394)(295,396)(296,395)(297,389)(298,390)(299,392)(300,391)(301,385)(302,386)(303,388)(304,387)(305,381)(306,382)(307,384)(308,383)(309,377)(310,378)(311,380)(312,379)(313,373)(314,374)(315,376)(316,375)(317,369)(318,370)(319,372)(320,371)(321,365)(322,366)(323,368)(324,367)(325,361)(326,362)(327,364)(328,363)(329,357)(330,358)(331,360)(332,359)(333,353)(334,354)(335,356)(336,355)(337,349)(338,350)(339,352)(340,351);;
s2 := (  1,273)(  2,276)(  3,275)(  4,274)(  5,277)(  6,280)(  7,279)(  8,278)(  9,281)( 10,284)( 11,283)( 12,282)( 13,285)( 14,288)( 15,287)( 16,286)( 17,289)( 18,292)( 19,291)( 20,290)( 21,293)( 22,296)( 23,295)( 24,294)( 25,297)( 26,300)( 27,299)( 28,298)( 29,301)( 30,304)( 31,303)( 32,302)( 33,305)( 34,308)( 35,307)( 36,306)( 37,309)( 38,312)( 39,311)( 40,310)( 41,313)( 42,316)( 43,315)( 44,314)( 45,317)( 46,320)( 47,319)( 48,318)( 49,321)( 50,324)( 51,323)( 52,322)( 53,325)( 54,328)( 55,327)( 56,326)( 57,329)( 58,332)( 59,331)( 60,330)( 61,333)( 62,336)( 63,335)( 64,334)( 65,337)( 66,340)( 67,339)( 68,338)( 69,205)( 70,208)( 71,207)( 72,206)( 73,209)( 74,212)( 75,211)( 76,210)( 77,213)( 78,216)( 79,215)( 80,214)( 81,217)( 82,220)( 83,219)( 84,218)( 85,221)( 86,224)( 87,223)( 88,222)( 89,225)( 90,228)( 91,227)( 92,226)( 93,229)( 94,232)( 95,231)( 96,230)( 97,233)( 98,236)( 99,235)(100,234)(101,237)(102,240)(103,239)(104,238)(105,241)(106,244)(107,243)(108,242)(109,245)(110,248)(111,247)(112,246)(113,249)(114,252)(115,251)(116,250)(117,253)(118,256)(119,255)(120,254)(121,257)(122,260)(123,259)(124,258)(125,261)(126,264)(127,263)(128,262)(129,265)(130,268)(131,267)(132,266)(133,269)(134,272)(135,271)(136,270)(137,341)(138,344)(139,343)(140,342)(141,345)(142,348)(143,347)(144,346)(145,349)(146,352)(147,351)(148,350)(149,353)(150,356)(151,355)(152,354)(153,357)(154,360)(155,359)(156,358)(157,361)(158,364)(159,363)(160,362)(161,365)(162,368)(163,367)(164,366)(165,369)(166,372)(167,371)(168,370)(169,373)(170,376)(171,375)(172,374)(173,377)(174,380)(175,379)(176,378)(177,381)(178,384)(179,383)(180,382)(181,385)(182,388)(183,387)(184,386)(185,389)(186,392)(187,391)(188,390)(189,393)(190,396)(191,395)(192,394)(193,397)(194,400)(195,399)(196,398)(197,401)(198,404)(199,403)(200,402)(201,405)(202,408)(203,407)(204,406);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(408)!(  1,  3)(  2,  4)(  5, 67)(  6, 68)(  7, 65)(  8, 66)(  9, 63)( 10, 64)( 11, 61)( 12, 62)( 13, 59)( 14, 60)( 15, 57)( 16, 58)( 17, 55)( 18, 56)( 19, 53)( 20, 54)( 21, 51)( 22, 52)( 23, 49)( 24, 50)( 25, 47)( 26, 48)( 27, 45)( 28, 46)( 29, 43)( 30, 44)( 31, 41)( 32, 42)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 69, 71)( 70, 72)( 73,135)( 74,136)( 75,133)( 76,134)( 77,131)( 78,132)( 79,129)( 80,130)( 81,127)( 82,128)( 83,125)( 84,126)( 85,123)( 86,124)( 87,121)( 88,122)( 89,119)( 90,120)( 91,117)( 92,118)( 93,115)( 94,116)( 95,113)( 96,114)( 97,111)( 98,112)( 99,109)(100,110)(101,107)(102,108)(103,105)(104,106)(137,139)(138,140)(141,203)(142,204)(143,201)(144,202)(145,199)(146,200)(147,197)(148,198)(149,195)(150,196)(151,193)(152,194)(153,191)(154,192)(155,189)(156,190)(157,187)(158,188)(159,185)(160,186)(161,183)(162,184)(163,181)(164,182)(165,179)(166,180)(167,177)(168,178)(169,175)(170,176)(171,173)(172,174)(205,207)(206,208)(209,271)(210,272)(211,269)(212,270)(213,267)(214,268)(215,265)(216,266)(217,263)(218,264)(219,261)(220,262)(221,259)(222,260)(223,257)(224,258)(225,255)(226,256)(227,253)(228,254)(229,251)(230,252)(231,249)(232,250)(233,247)(234,248)(235,245)(236,246)(237,243)(238,244)(239,241)(240,242)(273,275)(274,276)(277,339)(278,340)(279,337)(280,338)(281,335)(282,336)(283,333)(284,334)(285,331)(286,332)(287,329)(288,330)(289,327)(290,328)(291,325)(292,326)(293,323)(294,324)(295,321)(296,322)(297,319)(298,320)(299,317)(300,318)(301,315)(302,316)(303,313)(304,314)(305,311)(306,312)(307,309)(308,310)(341,343)(342,344)(345,407)(346,408)(347,405)(348,406)(349,403)(350,404)(351,401)(352,402)(353,399)(354,400)(355,397)(356,398)(357,395)(358,396)(359,393)(360,394)(361,391)(362,392)(363,389)(364,390)(365,387)(366,388)(367,385)(368,386)(369,383)(370,384)(371,381)(372,382)(373,379)(374,380)(375,377)(376,378);
s1 := Sym(408)!(  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 65)( 10, 66)( 11, 68)( 12, 67)( 13, 61)( 14, 62)( 15, 64)( 16, 63)( 17, 57)( 18, 58)( 19, 60)( 20, 59)( 21, 53)( 22, 54)( 23, 56)( 24, 55)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 33, 41)( 34, 42)( 35, 44)( 36, 43)( 39, 40)( 69,141)( 70,142)( 71,144)( 72,143)( 73,137)( 74,138)( 75,140)( 76,139)( 77,201)( 78,202)( 79,204)( 80,203)( 81,197)( 82,198)( 83,200)( 84,199)( 85,193)( 86,194)( 87,196)( 88,195)( 89,189)( 90,190)( 91,192)( 92,191)( 93,185)( 94,186)( 95,188)( 96,187)( 97,181)( 98,182)( 99,184)(100,183)(101,177)(102,178)(103,180)(104,179)(105,173)(106,174)(107,176)(108,175)(109,169)(110,170)(111,172)(112,171)(113,165)(114,166)(115,168)(116,167)(117,161)(118,162)(119,164)(120,163)(121,157)(122,158)(123,160)(124,159)(125,153)(126,154)(127,156)(128,155)(129,149)(130,150)(131,152)(132,151)(133,145)(134,146)(135,148)(136,147)(205,209)(206,210)(207,212)(208,211)(213,269)(214,270)(215,272)(216,271)(217,265)(218,266)(219,268)(220,267)(221,261)(222,262)(223,264)(224,263)(225,257)(226,258)(227,260)(228,259)(229,253)(230,254)(231,256)(232,255)(233,249)(234,250)(235,252)(236,251)(237,245)(238,246)(239,248)(240,247)(243,244)(273,345)(274,346)(275,348)(276,347)(277,341)(278,342)(279,344)(280,343)(281,405)(282,406)(283,408)(284,407)(285,401)(286,402)(287,404)(288,403)(289,397)(290,398)(291,400)(292,399)(293,393)(294,394)(295,396)(296,395)(297,389)(298,390)(299,392)(300,391)(301,385)(302,386)(303,388)(304,387)(305,381)(306,382)(307,384)(308,383)(309,377)(310,378)(311,380)(312,379)(313,373)(314,374)(315,376)(316,375)(317,369)(318,370)(319,372)(320,371)(321,365)(322,366)(323,368)(324,367)(325,361)(326,362)(327,364)(328,363)(329,357)(330,358)(331,360)(332,359)(333,353)(334,354)(335,356)(336,355)(337,349)(338,350)(339,352)(340,351);
s2 := Sym(408)!(  1,273)(  2,276)(  3,275)(  4,274)(  5,277)(  6,280)(  7,279)(  8,278)(  9,281)( 10,284)( 11,283)( 12,282)( 13,285)( 14,288)( 15,287)( 16,286)( 17,289)( 18,292)( 19,291)( 20,290)( 21,293)( 22,296)( 23,295)( 24,294)( 25,297)( 26,300)( 27,299)( 28,298)( 29,301)( 30,304)( 31,303)( 32,302)( 33,305)( 34,308)( 35,307)( 36,306)( 37,309)( 38,312)( 39,311)( 40,310)( 41,313)( 42,316)( 43,315)( 44,314)( 45,317)( 46,320)( 47,319)( 48,318)( 49,321)( 50,324)( 51,323)( 52,322)( 53,325)( 54,328)( 55,327)( 56,326)( 57,329)( 58,332)( 59,331)( 60,330)( 61,333)( 62,336)( 63,335)( 64,334)( 65,337)( 66,340)( 67,339)( 68,338)( 69,205)( 70,208)( 71,207)( 72,206)( 73,209)( 74,212)( 75,211)( 76,210)( 77,213)( 78,216)( 79,215)( 80,214)( 81,217)( 82,220)( 83,219)( 84,218)( 85,221)( 86,224)( 87,223)( 88,222)( 89,225)( 90,228)( 91,227)( 92,226)( 93,229)( 94,232)( 95,231)( 96,230)( 97,233)( 98,236)( 99,235)(100,234)(101,237)(102,240)(103,239)(104,238)(105,241)(106,244)(107,243)(108,242)(109,245)(110,248)(111,247)(112,246)(113,249)(114,252)(115,251)(116,250)(117,253)(118,256)(119,255)(120,254)(121,257)(122,260)(123,259)(124,258)(125,261)(126,264)(127,263)(128,262)(129,265)(130,268)(131,267)(132,266)(133,269)(134,272)(135,271)(136,270)(137,341)(138,344)(139,343)(140,342)(141,345)(142,348)(143,347)(144,346)(145,349)(146,352)(147,351)(148,350)(149,353)(150,356)(151,355)(152,354)(153,357)(154,360)(155,359)(156,358)(157,361)(158,364)(159,363)(160,362)(161,365)(162,368)(163,367)(164,366)(165,369)(166,372)(167,371)(168,370)(169,373)(170,376)(171,375)(172,374)(173,377)(174,380)(175,379)(176,378)(177,381)(178,384)(179,383)(180,382)(181,385)(182,388)(183,387)(184,386)(185,389)(186,392)(187,391)(188,390)(189,393)(190,396)(191,395)(192,394)(193,397)(194,400)(195,399)(196,398)(197,401)(198,404)(199,403)(200,402)(201,405)(202,408)(203,407)(204,406);
poly := sub<Sym(408)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0 >;
References : None.
to this polytope

Twisty Puzzle